On actions and strict actions in homological categories

Manfred Hartl and Bruno Loiseau

Let $G$ be an object of a finitely cocomplete homological category
$\mathbb C$. We study actions of $G$ on objects $A$ of $\mathbb C$
(defined by
Bourn and Janelidze as being algebras over a certain monad $\mathbb T_G$),
with two objectives: investigating to which extent actions can be
described in terms of smaller data, called action cores; and to single out
those abstract action cores which extend to actions corresponding to
semi-direct products of $A$ and $G$ (in a non-exact setting, not every
action does). This amounts to exhibiting a subcategory of the category of
the actions of $G$ on objects $A$ which is equivalent with the category of
points in $\mathbb C$ over $G$, and to describing it in terms of action
cores. This notion and its study are based on a preliminary investigation
of co-smash products, in which cross-effects of functors in a general
categorical context turn out to be a useful tool. The co-smash products
also allow us to define higher categorical commutators, different from the
ones of Huq, which are not generally expressible in terms of nested binary
ones. We use strict action cores to show that any normal subobject of an
object $E$ (i.e., the equivalence class of $0$ for some equivalence
relation on $E$ in $\mathbb C$) admits a strict conjugation action of $E$.
If $\mathbb C$ is semi-abelian, we show that for subobjects $X$, $Y$ of
some object $A$, $X$ is proper in the supremum of $X$ and $Y$ if and only
if $X$ is stable under the restriction to $Y$ of the conjugation action of
$A$ on itself. This also amounts to an alternative proof of Bourn and
Janelidze's category equivalence between points over $G$ in $\mathbb C$
and actions of $G$ in the semi-abelian context. Finally, we show that the
two axioms of an algebra which characterize $G$-actions are equivalent
with three others ones, in terms of action cores. These axioms are
commutative squares involving only co-smash products. Two of them are
associativity type conditions which generalize the usual properties of an
action of one group on another, while the third is kind of a higher
coherence condition which is a consequence of the other two in the
category of groups, but probably not in general. As an application, we
characterize abelian action cores, that is, action cores corresponding to
Beck modules; here also the coherence condition follows from the others.